The graphs of non-degenerate linear codes

Abstract

We consider the Grassmann graph of k-dimensional subspaces of an n-dimensional vector space over the q-element field and its subgraph Γ(n,k)q which consists of all non-degenerate linear [n,k]q codes. We assume that 1<k<n−1. It is well-known that every automorphism of the Grassmann graph is induced by a semilinear automorphism of the corresponding vector space or a semilinear isomorphism to the dual vector space and the second possibility is realized only for n=2k. Our result is the following: if q≥3 or k≠2, then every isomorphism of Γ(n,k)q to a subgraph of the Grassmann graph can be uniquely extended to an automorphism of the Grassmann graph; in the case when q=k=2, there are subgraphs of the Grassmann graph isomorphic to Γ(n,k)q and such that isomorphisms between these subgraphs and Γ(n,k)q cannot be extended to automorphisms of the Grassmann graph.